Astronomy and Gravity Tutorial




Key Concepts:

Gravity

Kepler's Law

Circular Motion

Planetary Orbits

Question 1:

"The mass of the Earth, the mass of the moon and the distance between them are


.

Determine the gravitational force of the moon upon the earth and the gravitational force of the earth upon the moon."

To solve this problem, we need:


  1. A sketch of the situation we are trying to model.

  2. An equation that tells us the gravitational force between two objects.

Let us start with 1.

1. A sketch of the situation we are trying to model.



Here we have defined the symbols:





Now let us see what we can do about 2.

2. An equation that tells us the gravitational force between two objects.

In this situation, can we use Fg=mg and just plug in the mass of the moon or the earth? It is tempting, but the answer is NO! The gravitational acceleration, g=9.82 m/s^2, is only useful when we are close to the surface of the earth. It is only true in that very specific case. In general, we have to use Newton's Law of Gravitation:




It basically says that:

"To find the gravitational force between two object, you have to multiply their masses, divide by the distance between them and multiply by a constant."

It makes sense that bigger masses means more force. It also makes sense that the further the masses get from each other, the weaker the force becomes. We can look up the Gravitational Constant, G, and plug in the numbers:





That looks pretty complicated. Let us try to clean the expression up before we reach for the calculator. Move all the "raw numbers" to the left and all the "powers of 10" to the right:




Much better! Let us take care of the powers of 10:



Now for the "raw numbers" in front:



So there is a pretty huge force between them! Notice that:

"Force on Earth from Moon = Force on Moon from Earth"

Question 2

"The use the information in Question 1 to determine how long it takes the moon to orbit the earth once."

Seriously? That seems downright impossible! But let us see what we would need in order to figure this out:

  1. A sketch of the situation we are trying to model.

  2. An equation that describes the force pulling the moon towards the earth.

  3. An equation that describes the force pulling the moon away from the earth.

  4. Some way of using this to figure out the orbit time of the moon.

Let us start with 1.

  1. A sketch of the situation we are trying to model.

We have the moon going in a circular orbit around the earth. We know the radius of this orbit and we know the masses of the two planets. We want to find the time, T, it takes to go around the earth once.

Now on to 2.

2. An equation that describes the force pulling the moon towards the earth,

We need an equation that gives us the gravitational force between the two planets. We can just reuse: